Solve simutaneously. x(√x) + y(√y) = 183 x(√y) + y(√x) = 185


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Question Number 13909 by tawa tawa last updated on 25/May/17

$Solve simutaneously .$ $x \sqrt{x} + y \sqrt{y} = 183$ $x \sqrt{y} + y \sqrt{x} = 185$
Commented by RasheedSindhi last updated on 25/May/17

${(x^{1 / 2})}^{3} + {(y^{1 / 2})}^{3} = 183$ $\Rightarrow (x^{1 / 2} + y^{1 / 2}) (x - x^{1 / 2} y^{1 / 2} + y) = 183 . . . . (i)$ $x^{1 / 2} y^{1 / 2} (x^{1 / 2} + y^{1 / 2}) = 185 . . . . (ii)$ $(i) / (ii) :$ $\frac{(x - x^{1 / 2} y^{1 / 2} + y)}{x^{1 / 2} y^{1 / 2}} = \frac{183}{185}$ $\frac{x^{1 / 2}}{y^{1 / 2}} - 1 + \frac{y^{1 / 2}}{x^{1 / 2}} = \frac{183}{185}$ $Let \frac{x^{1 / 2}}{y^{1 / 2}} = t$ $t + \frac{1}{t} = \frac{183 + 185}{185} = \frac{368}{185}$ $t^{2} + 1 + \frac{2 t}{185} = 0$ ${185 t}^{2} + 2 t - 185 = 0$ $t = \frac{- 2 \pm \sqrt{4 - 4 {(185)}^{2}}}{2 (185)}$ $t = \frac{- 2 \pm 2 i \sqrt{185^{2} - 1}}{2 (185)}$ $t = \frac{- 1 \pm i \sqrt{34224}}{185}$ $\frac{x^{1 / 2}}{y^{1 / 2}} = \frac{- 1 \pm 4 i \sqrt{2139}}{185}$ $\frac{x}{y} = {(\frac{- 1 \pm 4 i \sqrt{2139}}{185})}^{2} = a^{2}$ $x = a^{2} y$ $putting in (i)$ $a^{2} y \sqrt{a^{2} y} + y \sqrt{y} = 183$ $a^{3} y \sqrt{y} + y \sqrt{y} = 183$ $y^{3 / 2} = \frac{183}{a^{3} + 1}$ $y = {(\frac{183}{a^{3} + 1})}^{2 / 3}$ $x = a^{2} {(\frac{183}{a^{3} + 1})}^{2 / 3}$ $Where a = \frac{- 1 \pm 4 i \sqrt{2139}}{185}$ $Some mistkes are there in this$ $solution .$
Commented by ajfour last updated on 25/May/17

$s h o u l d h a v e b e e n e a s y i f :$ $x \sqrt{x} + y \sqrt{y} = 189$ $x \sqrt{y} + y \sqrt{x} = 180$
Commented by mrW1 last updated on 25/May/17

$Y o u a r e r i g h t . T h e g i v e n n u m b e r s$ $d e l i v e r n o r e a l s o l u t i o n a n d m a k e$ $i t d i f f i c u l t t o c h e c k t h e r e s u l t .$
Commented by ajfour last updated on 25/May/17

$S i r, i f y o u c a n s e e Q . 13903 . . i^{'} d b e$ $v e r y t h a n k f u l . .$
Commented by tawa tawa last updated on 25/May/17

$God bless you sir$
	Answered by mrW1 last updated on 25/May/17
	$u=(√x) v=(√y) u^3 +v^3 =183 (189) u^2 v+v^2 u=185 (180) (u+v)^3 =u^3 +v^3 +3(u^2 v+v^2 u)=183+3×185=738 (=189+3×180=729) u+v=^3 (√(738))=9.037=a (=^3 (√(729))=9) uv(u+v)=uva u^2 v+v^2 u=uva 185=uva (180=9uv) uv=((185)/a)=20.472=b (=((180)/9)=20) { ((u+v=a (9))),((uv=b (20))) :} (u−v)^2 =u^2 +v^2 −2uv=u^2 +v^2 +2uv−4uv =(u+v)^2 −4uv=a^2 −4b=−0.2206 (=9^2 −4×20=1) u−v=±0.47i=±ci (=±1) ⇒(√x)=u=(((u+v)/2))+(((u−v)/2))=(a/2)±(c/2)i=4.52±0.235i ((9/2)±(1/2)=5 or 4) ⇒(√y)=v=(((u+v)/2))−(((u−v)/2))=(a/2)∓(c/2)i=4.52∓0.235i ((9/2)∓(1/2)=4 or 5) ⇒x=(4.5±0.235i)^2 =20.375±2.124i (5^2 =25 or 4^2 =16) ⇒y=(4.5∓0.235i)^2 =20.375∓2.124i (4^2 =16 or 5^2 =25)$
	$u = \sqrt{x}$ $v = \sqrt{y}$ $u^{3} + v^{3} = 183 (189)$ $u^{2} v + v^{2} u = 185 (180)$ ${(u + v)}^{3} = u^{3} + v^{3} + 3 (u^{2} v + v^{2} u) = 183 + 3 \times 185 = 738 (= 189 + 3 \times 180 = 729)$ $u + v =^{3} \sqrt{738} = 9.037 = a (=^{3} \sqrt{729} = 9)$ $u v (u + v) = u v a$ $u^{2} v + v^{2} u = u v a$ $185 = u v a (180 = 9 u v)$ $u v = \frac{185}{a} = 20.472 = b (= \frac{180}{9} = 20)$ ${\begin{cases} u + v = a (9) \\ u v = b (20) \end{cases}$ ${(u - v)}^{2} = u^{2} + v^{2} - 2 u v = u^{2} + v^{2} + 2 u v - 4 u v$ $= {(u + v)}^{2} - 4 u v = a^{2} - 4 b = - 0.2206 (= 9^{2} - 4 \times 20 = 1)$ $u - v = \pm 0.47 i = \pm c i (= \pm 1)$ $\Rightarrow \sqrt{x} = u = (\frac{u + v}{2}) + (\frac{u - v}{2}) = \frac{a}{2} \pm \frac{c}{2} i = 4.52 \pm 0.235 i (\frac{9}{2} \pm \frac{1}{2} = 5 o r 4)$ $\Rightarrow \sqrt{y} = v = (\frac{u + v}{2}) - (\frac{u - v}{2}) = \frac{a}{2} \mp \frac{c}{2} i = 4.52 \mp 0.235 i (\frac{9}{2} \mp \frac{1}{2} = 4 o r 5)$ $\Rightarrow x = {(4.5 \pm 0.235 i)}^{2} = 20.375 \pm 2.124 i (5^{2} = 25 o r 4^{2} = 16)$ $\Rightarrow y = {(4.5 \mp 0.235 i)}^{2} = 20.375 \mp 2.124 i (4^{2} = 16 o r 5^{2} = 25)$
	Commented by RasheedSindhi last updated on 25/May/17

	$V N i c e S i r!$
	Commented by mrW1 last updated on 25/May/17

	$T h a n k y o u s i r!$
	Commented by tawa tawa last updated on 25/May/17

	$God bless you sir .$
	Answered by AH Soomro last updated on 25/May/17

	$A d i f f e r e n t t r y$ $x \sqrt{x} + y \sqrt{y} = 183 . . . . . . . . . . (i)$ $x \sqrt{y} + y \sqrt{x} = 185 . . . . . . . . . . (i i)$ ${(i)}^{2} - {(i i)}^{2} :$ $x^{3} + 2 xy \sqrt{xy} + y^{3} = 33489$ $x^{2} y + 2 xy \sqrt{xy} + {xy}^{2} = 34225$ $x^{3} + y^{3} - (x^{2} y + {xy}^{2}) = - 736$ $(x + y) (x^{2} - xy + y^{2}) - xy (x + y) = - 736$ $(x + y) {(x - y)}^{2} = - 736$ $Continue$
	Commented by tawa tawa last updated on 25/May/17

	$God bless you sir .$
	Answered by RasheedSindhi last updated on 25/May/17

	$Answer of the modified question .$ $Modification suggested by ajfour .$ $x \sqrt{x} + y \sqrt{y} = 189 . . . . . . . . . . . (i)$ $x \sqrt{y} + y \sqrt{x} = 180 . . . . . . . . . . . . (ii)$ $(i) \Rightarrow x^{3 / 2} + y^{3 / 2} = 189$ $\Rightarrow (x^{1 / 2} + y^{1 / 2}) (x - x^{1 / 2} y^{1 / 2} + y) = 189 . . . . (iii)$ $(ii) \Rightarrow x^{1 / 2} y^{1 / 2} (x^{1 / 2} + y^{1 / 2}) = 180 . . . (iv)$ $(iii) / (iv) : \frac{x - x^{1 / 2} y^{1 / 2} + y}{x^{1 / 2} y^{1 / 2}} = \frac{189}{180}$ $\frac{x^{1 / 2}}{y^{1 / 2}} - 1 + \frac{y^{1 / 2}}{x^{1 / 2}} = \frac{21}{20}$ $\frac{x^{1 / 2}}{y^{1 / 2}} + \frac{y^{1 / 2}}{x^{1 / 2}} = \frac{21}{20} + 1 = \frac{21 + 20}{20} = \frac{41}{20}$ $t + \frac{1}{t} = \frac{41}{20} [t = \frac{x^{1 / 2}}{y^{1 / 2}}]$ ${20 t}^{2} - 41 t + 20 = 0$ $t = \frac{5}{4}, \frac{4}{5}$ $\frac{x^{1 / 2}}{y^{1 / 2}} = \frac{5}{4}, \frac{4}{5}$ $\frac{x}{y} = \frac{25}{16}, \frac{16}{25}$ $x = \frac{25}{16} y, \frac{16}{25} y$ $putting these values in (i)$ $x \sqrt{x} + y \sqrt{y} = 189$ $\Rightarrow (\frac{25}{16} y) \sqrt{\frac{25}{16} y} + y \sqrt{y} = 189$ $\Rightarrow (\frac{16}{25} y) \sqrt{\frac{16}{25} y} + y \sqrt{y} = 189$ $y \sqrt{y} (\frac{125}{64} + 1) = 189$ $y \sqrt{y} (\frac{64}{125} + 1) = 189$ $y \sqrt{y} (\frac{189}{64}) = 189$ $y \sqrt{y} (\frac{189}{125}) = 189$ $y^{3 / 2} = 64$ $y^{3 / 2} = 125$ $y = 64^{2 / 3} = 16$ $y = 125^{2 / 3} = 25$ $x = \frac{25}{16} y = \frac{25}{16} (16) = 25$ $x = \frac{16}{25} y = \frac{16}{25} (25) = 16$ $(x, y) = (16, 25) or (25, 16)$
	Commented by tawa tawa last updated on 25/May/17

	$God bless you sir$
	Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/May/17

	$\sqrt{x} = t, \sqrt{y} = s$ $t^{3} + s^{3} = 183$ $t^{2} s + {t s}^{2} = 185 \Rightarrow t s (t + s) = 185$ ${(t + s)}^{3} = t^{3} + s^{3} + 3 t s (t + s) = 183 + 3 \times 185 =$ $\Rightarrow t + s = \sqrt[3]{738} = 9.03 \Rightarrow t s = \frac{185}{9.03} = 20.49$ $z^{2} - 9.03 z + 20.49 = 0 \Rightarrow z = \frac{9.03 \pm \sqrt{9 . 03^{2} - 4 \times 20.49}}{2}$ $\Rightarrow z = 4.51 \pm 0.32 i$ $\sqrt{x} = 4.51 \pm 0.32 i \Rightarrow x = 20.24 \pm 2.88 i$ $\sqrt{y} = 4.51 \pm 0.32 i \Rightarrow y = 20.24 \pm 2.88 i$
	Commented by tawa tawa last updated on 25/May/17

	$God bless you sir .$
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